arXiv research

Study Szegő kernel on non-compact CR manifolds with specific conditions.

New classification for certain compact manifolds with positive isotropic curvature.

The study classifies quasi-Einstein manifolds with boundary.

Method calculates function integrals on complex manifolds.

In this paper, we completely classify all compact 4-manifolds with positive isotropic curvature. We show that they are diffeomorphic to $\mathbb{S}^4,$ or $\mathbb{R}\mathbb{P}^4$ or quotients of $\mathbb{S}^3\times \mathbb{R}$ by a cocompact fixed point free subgroup of the isometry group of the standard metric of $\m…

The study resolves a conjecture about harmonic forms on compact manifolds.

Extends mapping results to non-compact Riemannian manifolds with positive reach.

We exhibit the first examples of compact orientable hyperbolic manifolds that do not have any spin structure. We show that such manifolds exist in all dimensions $n \geq 4$. The core of the argument is the construction of a compact orientable hyperbolic $4$-manifold $M$ that contains a surface $S$ of genus $3$ with sel…

Given a compact Riemannian manifold $M$ without boundary, we show that large isoperimetric regions in $M\times\mathbb{R}^k$ are tubular neighborhoods of $M\times\{x\}$, with $x\in\mathbb{R}^k$.

Unique shrinking gradient Kähler-Ricci solitons found on non-compact toric manifolds.

Compact hyperbolic complex manifolds are rigid under deformation.

The article defines and compares two types of quantizations on compact manifolds.

The renormalized volume is reinterpreted using isoperimetric profiles.

The paper investigates conditions for compactness of submanifolds in Kahler manifolds.

The purpose of this paper is to classify all compact manifolds modeled on the 4-dimensional solvable Lie group $Sol_1^4$. The maximal compact subgroup of $Isom(Sol_1^4)$ is $D_4=\mathbb Z_4\rtimes\mathbb Z_2$. We shall exhibit an infra-solvmanifold with $Sol_1^4$-geometry whose holonomy is $D_4$. This implies that all …

The paper studies heat kernel asymptotics for Kohn Laplacians on CR manifolds.

Study functionals on almost complex structures for Yau's Challenge.

Compactness theorem for 3-manifold Floer theory defined by Fueter sections.

Study of Sasakian immersions in Heisenberg group and $\mathbb{B}^N imes \mathbb{R}$, focusing on $η$-Einstein case.

In this paper, we introduce the notion of maximal actions of compact tori on smooth manifolds and study compact connected complex manifolds equipped with maximal actions of compact tori. We give a complete classification of such manifolds, in terms of combinatorial objects, which are triples $(Δ, \mathfrak{h}, G)$ of n…

In this note we prove that a four-dimensional compact oriented half-confor\-mally flat Riemannian manifold $M^4$ is topologically $\mathbb{S}^{4}$ or $\mathbb{C}\mathbb{P}^{2},$ provided that the sectional curvatures all lie in the interval $[\frac{3\sqrt{3}-5}{4},\,1].$ In addition, we use the notion of biorthogonal (…

We study the possibility of realizing exotic smooth structures on punctured simply connected $4$-manifolds as leaves of a codimension one foliation on a compact manifold. In particular, we show the existence of uncountably many smooth open $4$-manifolds which are not diffeomorphic to any leaf of a codimension one trans…

The paper finds closed hypersurfaces with prescribed mean curvature in non-compact manifolds.

Conditions for scalar curvature on compact manifolds under conformal deformation.

Inspired by the role geometric structures play in our understanding of surfaces and three-manifolds, and Berger's observation that a surface of constant sectional curvature is determined up to local isometry by its Laplace spectrum, we explore the extent to which compact locally homogeneous three-manifolds are characte…

The paper constructs monopole Floer homology for specific 3-manifolds and surfaces.

Open manifolds can be covered by $\mathbb{R}^m$ with finite or infinite degree.

Let $M^n$ be a compact Kähler manifold with bisectional curvature bounded from below by $1$. If $diam(M) = π/ \sqrt{2}$ and $vol(M)> vol(\mathbb{C}\mathbb{P}^n)/ 2^n$, we prove that $M$ is biholomorphically isometric to $\mathbb{C}\mathbb{P}^n$ with the standard Fubini-Study metric.

New compact manifolds with closed G2 structures found.

Theorem A. Let $M^n$ denote a closed Riemannian manifold with nonpositive sectional curvature and let $\tilde M^n$ be the universal cover of $M^n$ with the lifted metric. Suppose that the universal cover $\tilde M^n$ contains no totally geodesic embedded Euclidean plane $\mathbb{R}^2$ (i.e., $M^n$ is a visibility manif…

We classify compact conformally flat $n$-dimensional manifolds with constant positive scalar curvature and satisfying an optimal integral pinching condition: they are covered isometrically by either $\mathbb{S}^{n}$ with the round metric, $\mathbb{S}^{1}\times \mathbb{S}^{n-1}$ with the product metric or $\mathbb{S}^{1…

Study compact quasi-Einstein manifolds with boundary estimates.

We prove that a critical metric of the volume functional on a $4$-dimensional compact manifold with boundary satisfying a second-order vanishing condition on the Weyl tensor must be isometric to a geodesic ball in a simply connected space form $\mathbb{R}^{4}$, $\mathbb{H}^{4}$ or $\mathbb{S}^{4}.$ Moreover, we provide…

Study finds topological restrictions on 4-manifolds with uniformly positive scalar curvature.

Compactness theory for biharmonic maps on degenerating Einstein manifolds.

Conic singular sub-manifolds are Lipschitz Normally Embedded in compact non-Euclidean manifolds.

Classifies holonomy groups of Riemannian manifolds and finds compact ones imply cone structures.

3D hyperbolic manifolds map one-to-one to their boundary character varieties.

We classify the isoparametric functions on $\mathbb{R}^n\times\mathbb{M}^m$, $n, m\geq2$, with compact level sets, where $\mathbb{M}^m$ is a connected, closed Riemannian manifold of dimension $m$. Also, we classify the isoparametric hypersurfaces in $\mathbb{S}^2\times\mathbb{R}^2$ with constant principal curvatures.

Using the implicit function theorem, we prove existence of solutions of the so-called conformally covariant split system on compact 3-dimensional Riemannian manifolds. They give rise to non-Constant Mean Curvature (non-CMC) vacuum initial data for the Einstein equations. We investigate the conformally covariant split s…

Affine connections linked to Riccati distributions on compact surfaces.

Let $Y$ be a closed $3$-manifold such that all flat $SU(2)$-connections on $Y$ are $non$-$degenerate$. In this article, we prove a Uhlenbeck-type compactness theorem on $Y$ for stable flat $SL(2,\mathbb{C})$ connections satisfying an $L^{2}$-bound for the real curvature. Combining the compactness theorem and a previous…

The abstract discusses compact quotients of Riemannian products by discrete subgroups, generalizing Inoue-Bombieri surfaces.

The paper builds complex hyperbolic 2-manifolds with isolated singularities.

In this short note, using Siu-Yau's method [14], we give a new proof that any n-dimensional compact Kahler manifold with positive orthogonal bisectional curvature must be biholomorphic to $\mathbb{P}^n$.

We prove the following result: Let $(M,g_0)$ be a compact manifold of dimension $n\geq 12$ with positive isotropic curvature. Then $M$ is diffeomorphic to a spherical space form, or the total space of an orbifiber bundle over $\mathbb{S}^1$ or $\mathcal{I}$ with generic fiber diffeomorphic to $\mathbb{S}^{n-1}/Γ$ such …

Study eigenvalues and functions on specific Heisenberg manifolds.

The paper extends a connected-sum inequality to calculate λ-Yamabe invariants of certain manifolds.